455:
217:-like region of space containing a magnetic field such that its sides are everywhere parallel to the field. Consequently, the magnetic flux through these sides is zero, and the cross-sections along the tube's length have constant, equal magnetic flux. In the limit of a large magnetic Reynolds number, Alfvén's theorem requires that these surfaces of constant flux move with the fluid that they are embedded in. As such, magnetic flux tubes are frozen into the fluid.
167:
1602:
2723:
produces results equivalent to that of classical Alfvén's theorem under ideal conditions, while also describing flux conservation and magnetic reconnection under non-ideal (real-world) conditions. Thus stochastic flux-freezing solutions can provide better descriptions of observed phenomena without relying on idealized conditions that are rare or even absent in the observed environment.
1340:
1375:
1023:
744:
1761:
1952:
1131:
2898:
is a linear combination of the mean motions of the individual species weighted by the species' respective mass. Under Alfvén's theorem, the magnetic field is restricted to move with this bulk velocity, but not necessarily with the velocity of the individual species. As such, Alfvén's theorem does not
220:
The intersection of the sides of two magnetic flux tubes form a magnetic field line, a curve that is everywhere parallel to the magnetic field. In fluids where flux tubes are frozen-in, it then follows that magnetic field lines must also be frozen-in. However, the conditions for frozen-in field lines
2722:
Research in the 21st century has claimed that the classical Alfvén theorem is inconsistent with the phenomenon of spontaneous stochasticity. Stochastic conservation laws developed to describe hydrodynamic behavior are shown to apply in the magnetohydrodynamic regime as well. Using the same tools
2402:
2300:), the conditions for the latter are weaker than the conditions for the former. Unlike the conditions for flux conservation, the conditions for field line conservation can be satisfied when an additional, source term parallel to the magnetic field is present in the ideal induction equation.
2138:
2545:
Alfvén's theorem indicates that the magnetic field topology cannot change in a perfectly conducting fluid. However, in the case of complicated or turbulent flows, this would lead to highly tangled magnetic fields with very complicated topologies that should impede the fluid motions.
1597:{\displaystyle {\begin{aligned}{\frac {D\Phi _{B}}{Dt}}=\lim _{\delta t\to 0}\iint _{S_{1}}{\frac {\mathbf {B} (t+\delta t)-\mathbf {B} (t)}{\delta t}}\cdot d\mathbf {S} _{1}-\oint _{\partial S_{1}}\left(\mathbf {v} \times \mathbf {B} (t)\right)\cdot d\mathbf {l} .\end{aligned}}}
2689:
819:
2519:
556:
1617:
442:
2041:
148:. Magnetic flux conservation implies that the magnetic flux through a surface moving with the bulk fluid velocity is constant, and magnetic field line conservation implies that, if two fluid elements are connected by a magnetic field line, they will always be.
1840:
2262:
1335:{\displaystyle \iint _{S_{2}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{2}=\iint _{S_{1}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{1}-\oint _{\partial S_{1}}\left(\mathbf {v} \ \delta t\times \mathbf {B} (t)\right)\cdot d\mathbf {l} ,}
2899:
guarantee that individual species within the fluid will be restricted to move with the magnetic field, and currents can flow perpendicular to the magnetic field provided the bulk velocity matches the velocity of the magnetic field.
2833:
2309:
2429:
are frozen to the fluid, analogous to how magnetic flux tubes moving with a perfectly conducting ideal-MHD fluid are frozen to the fluid. The ideal induction equation takes the same form as the equation for vorticity
136:
are infinitely large. Motions of the two are constrained in that all bulk fluid motions perpendicular to the magnetic field result in matching perpendicular motion of the field at the same velocity and vice versa.
2052:
2580:
1018:{\displaystyle 0=-\iint _{S_{1}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{1}+\iint _{S_{2}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{2}+\iint _{S_{3}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{3},}
2454:
739:{\displaystyle {\frac {D\Phi _{B}}{Dt}}=\lim _{\delta t\to 0}{\frac {\iint _{S_{2}}\mathbf {B} (t+\delta t)\cdot d\mathbf {S} _{2}-\iint _{S_{1}}\mathbf {B} (t)\cdot d\mathbf {S} _{1}}{\delta t}}.}
1756:{\displaystyle {\frac {D\Phi _{B}}{Dt}}=\iint _{S_{1}}\left({\frac {\partial \mathbf {B} }{\partial t}}-\nabla \times \left(\mathbf {v} \times \mathbf {B} \right)\right)\cdot d\mathbf {S} _{1}.}
1380:
373:
1963:
1817:
315:
447:
The conservation of magnetic flux through material surfaces embedded in the fluid follows directly from the ideal induction equation and the assumption of no magnetic monopoles through
96:"On the Existence of Electromagnetic-Hydrodynamic Waves" interpreted the results of Alfvén's earlier paper "Existence of Electromagnetic-Hydrodynamic Waves", published in the journal
1947:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=(\mathbf {B} \cdot \nabla )\mathbf {v} -(\mathbf {v} \cdot \nabla )\mathbf {B} -\mathbf {B} (\nabla \cdot \mathbf {v} ).}
88:
In view of the infinite conductivity, every motion (perpendicular to the field) of the liquid in relation to the lines of force is forbidden because it would give infinite
2197:
3175:
Eyink, Gregory L.; Aluie, Hussein (November 2006). "The breakdown of Alfvén's theorem in ideal plasma flows: Necessary conditions and physical conjectures".
229:
In mathematical terms, Alfvén's theorem states that, in an electrically conducting fluid in the limit of a large magnetic
Reynolds number, the magnetic flux
1827:
Field line conservation can also be derived mathematically using the ideal induction equation, Gauss's law for magnetism, and the mass continuity equation.
140:
Formally, the connection between the movement of the fluid and the movement of the magnetic field is detailed in two primary results, often referred to as
2397:{\displaystyle \left({\frac {\partial \mathbf {B} }{\partial t}}-\nabla \times \left(\mathbf {v} \times \mathbf {B} \right)\right)\times \mathbf {B} =0,}
2744:
2921:
2558:. In fact, a very high electrical conductivity translates into high magnetic Reynolds numbers, which indicates that the plasma will be turbulent.
3266:
Eyink, Gregory; Aluie, Hussein (2006). "The breakdown of Alfvén's theorem in ideal plasma flows: Necessary conditions and physical conjectures".
3139:
2133:{\displaystyle {\frac {D}{Dt}}\left({\frac {\mathbf {B} }{\rho }}\right)=\left({\frac {\mathbf {B} }{\rho }}\cdot \nabla \right)\mathbf {v} .}
2854:.) The many virtual field-vectors that arrive at the same final point must be averaged to obtain the physical magnetic field at that point.
2684:{\displaystyle \nabla \times ({\bf {{w}\times {\bf {{B})=\eta \nabla ^{2}{\bf {{B}+\nabla \times ({\bf {{v}\times {\bf {{B}),}}}}}}}}}}}
80:
2554:
seems to occur in these plasmas unlike what would be expected from the flux freezing conditions. This has important implications for
367:
is then assumed to be small relative to the induction term and is neglected. The induction equation then reduces to its ideal form:
2514:{\displaystyle {\frac {\partial {\boldsymbol {\omega }}}{\partial t}}=\nabla \times (\mathbf {v} \times {\boldsymbol {\omega }}).}
758:
can be re expressed by applying Gauss's law for magnetism to assume that the magnetic flux through a closed surface formed by
3250:
3041:
2407:
whereas, for flux to be conserved, the fluid must satisfy the stronger condition imposed by the ideal induction equation.
117:
that electrically conducting fluids and the magnetic fields within are constrained to move together in the limit of large
3152:
437:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times \left(\mathbf {v} \times \mathbf {B} \right).}
2036:{\displaystyle {\frac {\partial \rho }{\partial t}}+(\mathbf {v} \cdot \nabla )\rho =-\rho \nabla \cdot \mathbf {v} ,}
3069:
3415:
Eyink, Gregory (2009). "Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models".
2731:
2416:
1772:
267:
3085:
Blackman, Eric G (1 March 2013). "On deriving flux freezing in magnetohydrodynamics by direct differentiation".
3521:
54:
of a fluid in the limit of a large magnetic
Reynolds number cannot change. This approximation breaks down in
32:
3319:
Wilmot-Smith, A. L.; Priest, E. R.; Horing, G. (2005). "Magnetic diffusion and the motion of field lines".
448:
78:
in a 1943 paper titled "On the
Existence of Electromagnetic-Hydrodynamic Waves", published in the journal
1029:
352:
243:
20:
356:
221:
are weaker than the conditions for frozen-in flux tubes, or, equivalently, for conservation of flux.
114:
3232:
2422:
210:
Alfvén's theorem is frequently expressed in terms of magnetic flux tubes and magnetic field lines.
118:
40:
2257:{\displaystyle {\frac {D\delta \mathbf {l} }{Dt}}=(\delta \mathbf {l} \cdot \nabla )\mathbf {v} }
239:
71:
454:
2726:
This generalized theorem states that magnetic field lines of the fine-grained magnetic field
2567:
2551:
2524:
However, the induction equation is linear, whereas there is a nonlinear relationship between
1607:
Applying the definition of a partial derivative to the integrand of the first term, applying
1346:
59:
3479:
3381:
3328:
3285:
3194:
3104:
2998:
2951:
2550:
with high electrical conductivities do not generally show such complicated tangled fields.
2547:
340:
3452:
Lalescu, Cristian C.; Shi, Yi-Kang; Eyink, Gregory; Drivas, Theodore D.; Vishniac, Ethan;
8:
2707:
1350:
3483:
3385:
3332:
3289:
3198:
3108:
3002:
2955:
3469:
3453:
3424:
3371:
3344:
3301:
3275:
3210:
3184:
3120:
3094:
2967:
1608:
364:
360:
3497:
3458:"Inertial-Range Reconnection in Magnetohydrodynamic Turbulence and in the Solar Wind"
3397:
3246:
3148:
3124:
3116:
3065:
3037:
2868:
2735:
532:. The rate of change of the magnetic flux through the surface as it is advected from
363:
at the velocity and length scales being studied. The diffusion term in the governing
129:
51:
3348:
3305:
3214:
2171:
is the velocity at the other end, the differential velocity between the two ends is
488:
In an electrically conducting fluid with a space- and time-dependent magnetic field
3492:
3487:
3457:
3434:
3389:
3336:
3293:
3238:
3202:
3112:
3029:
3006:
2971:
2959:
2873:
246:
97:
75:
44:
2828:{\displaystyle d{\bf {x}}={\bf {u}}({\bf {x}},t)dt+{\sqrt {2\eta }}d{\bf {W}}(t)}
1831:
3297:
3206:
3033:
3393:
2851:
2706:
has been used. Although, in some cases, this velocity field can be found using
253:
36:
3340:
3242:
2863:
3515:
2571:
2555:
55:
3010:
3501:
3401:
3028:. Astrophysics and Space Science Library. Vol. 427. pp. 101–142.
2711:
133:
89:
2986:
2942:
2847:
2426:
3280:
3189:
2303:
Mathematically, for field lines to be frozen-in, the fluid must satisfy
1611:
to the second term, and combining the resultant surface integrals gives
3362:
Eyink, Gregory (2011). "Stochastic flux freezing and magnetic dynamo".
3024:
Priest, E. (2016). "MHD Structures in Three-Dimensional
Reconnection".
1053:
points outwards from the enclosed volume. In the surface integral over
161:
3438:
188:
are cross sections of a magnetic flux tube; the magnetic flux through
92:. Thus the matter of the liquid is "fastened" to the lines of force...
70:
The concept of magnetic fields being frozen into fluids with infinite
2963:
2730:
are "frozen-in" to the stochastic trajectories solving the following
157:
166:
3474:
249:
214:
3429:
3376:
3099:
2570:
is not infinite, a similar result can be obtained by defining the
105:
Later in life, Alfvén advised against the use of his own theorem.
113:
Informally, Alfvén's theorem refers to the fundamental result in
2268:
which has the same form as the equation obtained previously for
1766:
Using the ideal induction equation, the integrand vanishes, and
2046:
the ideal induction equation can be further rearranged to give
2296:
While flux conservation implies field line conservation (see
1345:
where the final term was rewritten using the properties of
3318:
2922:"On the Existence of Electromagnetic-Hydrodynamic Waves"
2892:
In magnetohydrodynamics (MHD), the bulk velocity field
39:
are constrained to move together in the limit of large
3231:
Gubbins, David; Herrero-Bervera, Emilio, eds. (2007).
3230:
2987:"On frozen-in field lines and field-line reconnection"
1830:
The ideal induction equation can be rewritten using a
3451:
2747:
2583:
2457:
2312:
2200:
2055:
1966:
1843:
1775:
1620:
1378:
1353:
was taken. Substituting this into the expression for
1134:
822:
559:
376:
270:
2293:are initially parallel, they will remain parallel.
3062:Magnetic Reconnection: MHD Theory and Applications
2827:
2683:
2513:
2396:
2256:
2132:
2035:
1946:
1811:
1755:
1596:
1334:
1017:
738:
436:
309:
2665:
2645:
2610:
3513:
2710:equations, the existence and uniqueness of this
1414:
591:
3234:Encyclopedia of Geomagnetism and Paleomagnetism
2410:
2297:
3321:Geophysical & Astrophysical Fluid Dynamics
3147:. Taipei: Airiti Press Inc. pp. 173–176.
3064:(First ed.). Cambridge University Press.
151:
3226:
3224:
3059:
2717:
252:by a macroscopic, space- and time-dependent
2158:is the bulk plasma velocity at one end and
1812:{\displaystyle {\frac {D\Phi _{B}}{Dt}}=0.}
310:{\displaystyle {\frac {D\Phi _{B}}{Dt}}=0,}
3408:
3355:
3265:
3174:
3055:
3053:
2566:Even for the non-ideal case, in which the
1822:
3491:
3473:
3428:
3375:
3279:
3221:
3188:
3098:
2421:Kelvin's circulation theorem states that
500:, an arbitrary, orientable, open surface
224:
3084:
2929:Arkiv för matematik, astronomi och fysik
1116:. Solving for the surface integral over
1097:is the line element around the boundary
813:, this relationship can be expressed as
453:
165:
81:Arkiv för matematik, astronomi och fysik
3050:
2913:
2501:
2465:
3514:
3237:. Dordrecht: Springer. pp. 7–11.
3168:
3023:
2984:
2941:
2919:
2714:depends on the underlying conditions.
197:is equal to the magnetic flux through
3414:
3361:
3060:Priest, Eric; Forbes, Terry (2000).
2879:
2694:in which, instead of fluid velocity
1957:Using the mass continuity equation,
346:
47:, who put the idea forward in 1943.
3137:
2561:
1062:, the differential surface element
213:A magnetic flux tube is a tube- or
13:
2886:
2846:is the three-dimensional Gaussian
2639:
2620:
2584:
2574:transporting velocity by writing:
2483:
2471:
2461:
2343:
2331:
2321:
2243:
2114:
2019:
2001:
1978:
1970:
1927:
1905:
1880:
1857:
1847:
1783:
1698:
1686:
1676:
1628:
1529:
1390:
1262:
567:
402:
390:
380:
278:
50:Alfvén's theorem implies that the
14:
3533:
2298:§ Flux tubes and field lines
1834:and Gauss's law for magnetism as
2811:
2773:
2763:
2753:
2732:stochastic differential equation
2668:
2661:
2651:
2636:
2632:
2624:
2613:
2606:
2596:
2493:
2381:
2363:
2355:
2325:
2250:
2236:
2211:
2123:
2102:
2078:
2026:
1994:
1934:
1920:
1912:
1898:
1887:
1873:
1851:
1740:
1718:
1710:
1680:
1583:
1558:
1550:
1511:
1479:
1453:
1325:
1300:
1283:
1244:
1214:
1183:
1153:
1002:
972:
941:
911:
880:
850:
785:that connects the boundaries of
712:
691:
660:
630:
422:
414:
384:
146:magnetic field line conservation
115:ideal magnetohydrodynamic theory
3445:
3417:Journal of Mathematical Physics
3312:
3259:
3141:Elementary Space Plasma Physics
2991:Journal of Geophysical Research
2540:
16:Theorem in magnetohydrodynamics
3493:10.1103/PhysRevLett.115.025001
3268:Physica D: Nonlinear Phenomena
3177:Physica D: Nonlinear Phenomena
3131:
3078:
3017:
2978:
2935:
2822:
2816:
2784:
2768:
2590:
2505:
2489:
2246:
2229:
2143:Similarly, for a line segment
2004:
1990:
1938:
1924:
1908:
1894:
1883:
1869:
1568:
1562:
1489:
1483:
1472:
1457:
1424:
1310:
1304:
1233:
1218:
1172:
1157:
991:
976:
930:
915:
869:
854:
701:
695:
649:
634:
601:
33:electrically conducting fluids
1:
2906:
458:The closed surface formed by
128:—such as when the fluid is a
2842:is magnetic diffusivity and
2417:Kelvin's circulation theorem
2411:Kelvin's circulation theorem
7:
3298:10.1016/j.physd.2006.08.009
3207:10.1016/j.physd.2006.08.009
3087:European Journal of Physics
3034:10.1007/978-3-319-26432-5_3
2857:
2537:in the vorticity equation.
1369:and simplifying results in
119:magnetic Reynolds numbers (
108:
10:
3538:
3394:10.1103/PhysRevE.83.056405
3117:10.1088/0143-0807/34/2/489
2985:Alfvén, H. (August 1976).
2414:
749:The surface integral over
353:ideal magnetohydrodynamics
155:
152:Flux tubes and field lines
142:magnetic flux conservation
65:
21:ideal magnetohydrodynamics
3341:10.1080/03091920500044808
3243:10.1007/978-1-4020-4423-6
2718:Stochastic Alfvén theorem
1351:first-order approximation
449:Gauss's law for magnetism
41:magnetic Reynolds numbers
3138:Lyu, Ling-Hsiao (2010).
2442:in an ideal fluid where
3462:Physical Review Letters
3011:10.1029/JA081i022p04019
2920:Alfvén, Hannes (1943).
2448:is the velocity field:
1823:Field line conservation
72:electrical conductivity
2829:
2685:
2515:
2398:
2258:
2134:
2037:
1948:
1813:
1757:
1598:
1347:scalar triple products
1336:
1019:
740:
485:
438:
311:
225:Mathematical statement
207:
94:
74:was first proposed by
29:frozen-in flux theorem
3026:Magnetic Reconnection
2830:
2686:
2568:electric conductivity
2552:Magnetic reconnection
2548:Astrophysical plasmas
2516:
2415:Further information:
2399:
2259:
2135:
2038:
1949:
1814:
1758:
1599:
1337:
1041:was reversed so that
1020:
741:
457:
439:
312:
169:
156:Further information:
132:or when velocity and
86:
60:magnetic reconnection
3522:Magnetohydrodynamics
2745:
2700:, the flux velocity
2581:
2455:
2310:
2198:
2053:
1964:
1841:
1773:
1618:
1376:
1132:
820:
557:
374:
341:advective derivative
268:
43:. It is named after
3484:2015PhRvL.115b5001L
3386:2011PhRvE..83e6405E
3333:2005GApFD..99..177W
3290:2006PhyD..223...82E
3199:2006PhyD..223...82E
3109:2013EJPh...34..489B
3003:1976JGR....81.4019A
2956:1942Natur.150..405A
2708:magnetohydrodynamic
494:and velocity field
2825:
2681:
2511:
2394:
2254:
2130:
2033:
1944:
1809:
1753:
1594:
1592:
1431:
1332:
1015:
776:, and the surface
736:
608:
486:
434:
365:induction equation
361:magnetic diffusion
357:magnetic induction
307:
208:
3439:10.1063/1.3193681
3364:Physical Review E
3252:978-1-4020-3992-8
3043:978-3-319-26430-1
2997:(22): 4019–4021.
2880:Explanatory notes
2869:Magnetic pressure
2804:
2736:Langevin equation
2478:
2437:= ∇ ×
2338:
2224:
2109:
2085:
2069:
1985:
1864:
1801:
1693:
1646:
1501:
1413:
1408:
1289:
803:is zero. At time
731:
590:
585:
397:
347:Flux conservation
329:= ∂/∂
296:
130:perfect conductor
52:magnetic topology
3529:
3506:
3505:
3495:
3477:
3449:
3443:
3442:
3432:
3412:
3406:
3405:
3379:
3359:
3353:
3352:
3316:
3310:
3309:
3283:
3263:
3257:
3256:
3228:
3219:
3218:
3192:
3172:
3166:
3165:
3163:
3161:
3146:
3135:
3129:
3128:
3102:
3082:
3076:
3075:
3057:
3048:
3047:
3021:
3015:
3014:
2982:
2976:
2975:
2964:10.1038/150405d0
2939:
2933:
2932:
2926:
2917:
2900:
2897:
2890:
2874:Magnetic tension
2845:
2841:
2834:
2832:
2831:
2826:
2815:
2814:
2805:
2797:
2777:
2776:
2767:
2766:
2757:
2756:
2729:
2705:
2699:
2690:
2688:
2687:
2682:
2680:
2679:
2678:
2677:
2676:
2675:
2674:
2673:
2672:
2671:
2664:
2654:
2635:
2628:
2627:
2609:
2599:
2562:Resistive fluids
2556:magnetic dynamos
2536:
2530:
2520:
2518:
2517:
2512:
2504:
2496:
2479:
2477:
2469:
2468:
2459:
2447:
2441:
2403:
2401:
2400:
2395:
2384:
2376:
2372:
2371:
2367:
2366:
2358:
2339:
2337:
2329:
2328:
2319:
2292:
2286:
2278:. Therefore, if
2277:
2263:
2261:
2260:
2255:
2253:
2239:
2225:
2223:
2215:
2214:
2202:
2190:
2186:⋅ ∇)
2170:
2157:
2151:
2139:
2137:
2136:
2131:
2126:
2121:
2117:
2110:
2105:
2100:
2090:
2086:
2081:
2076:
2070:
2068:
2057:
2042:
2040:
2039:
2034:
2029:
1997:
1986:
1984:
1976:
1968:
1953:
1951:
1950:
1945:
1937:
1923:
1915:
1901:
1890:
1876:
1865:
1863:
1855:
1854:
1845:
1818:
1816:
1815:
1810:
1802:
1800:
1792:
1791:
1790:
1777:
1762:
1760:
1759:
1754:
1749:
1748:
1743:
1731:
1727:
1726:
1722:
1721:
1713:
1694:
1692:
1684:
1683:
1674:
1667:
1666:
1665:
1664:
1647:
1645:
1637:
1636:
1635:
1622:
1603:
1601:
1600:
1595:
1593:
1586:
1575:
1571:
1561:
1553:
1543:
1542:
1541:
1540:
1520:
1519:
1514:
1502:
1500:
1492:
1482:
1456:
1450:
1448:
1447:
1446:
1445:
1430:
1409:
1407:
1399:
1398:
1397:
1384:
1368:
1341:
1339:
1338:
1333:
1328:
1317:
1313:
1303:
1287:
1286:
1276:
1275:
1274:
1273:
1253:
1252:
1247:
1217:
1212:
1211:
1210:
1209:
1192:
1191:
1186:
1156:
1151:
1150:
1149:
1148:
1124:
1115:
1106:
1096:
1087:
1061:
1052:
1040:
1024:
1022:
1021:
1016:
1011:
1010:
1005:
975:
970:
969:
968:
967:
950:
949:
944:
914:
909:
908:
907:
906:
889:
888:
883:
853:
848:
847:
846:
845:
812:
802:
793:
784:
775:
766:
757:
745:
743:
742:
737:
732:
730:
722:
721:
720:
715:
694:
689:
688:
687:
686:
669:
668:
663:
633:
628:
627:
626:
625:
610:
607:
586:
584:
576:
575:
574:
561:
549:
540:
531:
522:
519:in a small time
518:
512:
508:
499:
493:
484:
475:
466:
443:
441:
440:
435:
430:
426:
425:
417:
398:
396:
388:
387:
378:
338:
337:⋅ ∇)
316:
314:
313:
308:
297:
295:
287:
286:
285:
272:
261:is constant, or
260:
247:material surface
237:
205:
196:
187:
178:
25:Alfvén's theorem
3537:
3536:
3532:
3531:
3530:
3528:
3527:
3526:
3512:
3511:
3510:
3509:
3450:
3446:
3413:
3409:
3360:
3356:
3317:
3313:
3281:physics/0607073
3264:
3260:
3253:
3229:
3222:
3190:physics/0607073
3173:
3169:
3159:
3157:
3155:
3144:
3136:
3132:
3083:
3079:
3072:
3058:
3051:
3044:
3022:
3018:
2983:
2979:
2940:
2936:
2924:
2918:
2914:
2909:
2904:
2903:
2893:
2891:
2887:
2882:
2860:
2843:
2839:
2810:
2809:
2796:
2772:
2771:
2762:
2761:
2752:
2751:
2746:
2743:
2742:
2734:, known as the
2727:
2720:
2701:
2695:
2660:
2659:
2658:
2650:
2649:
2648:
2631:
2630:
2629:
2623:
2619:
2605:
2604:
2603:
2595:
2594:
2593:
2582:
2579:
2578:
2564:
2543:
2532:
2526:∇ ×
2525:
2500:
2492:
2470:
2464:
2460:
2458:
2456:
2453:
2452:
2443:
2431:
2425:moving with an
2419:
2413:
2380:
2362:
2354:
2353:
2349:
2330:
2324:
2320:
2318:
2317:
2313:
2311:
2308:
2307:
2288:
2279:
2269:
2249:
2235:
2216:
2210:
2203:
2201:
2199:
2196:
2195:
2172:
2159:
2153:
2144:
2122:
2101:
2099:
2098:
2094:
2077:
2075:
2071:
2061:
2056:
2054:
2051:
2050:
2025:
1993:
1977:
1969:
1967:
1965:
1962:
1961:
1933:
1919:
1911:
1897:
1886:
1872:
1856:
1850:
1846:
1844:
1842:
1839:
1838:
1832:vector identity
1825:
1793:
1786:
1782:
1778:
1776:
1774:
1771:
1770:
1744:
1739:
1738:
1717:
1709:
1708:
1704:
1685:
1679:
1675:
1673:
1672:
1668:
1660:
1656:
1655:
1651:
1638:
1631:
1627:
1623:
1621:
1619:
1616:
1615:
1609:Stokes' theorem
1591:
1590:
1582:
1557:
1549:
1548:
1544:
1536:
1532:
1528:
1524:
1515:
1510:
1509:
1493:
1478:
1452:
1451:
1449:
1441:
1437:
1436:
1432:
1417:
1400:
1393:
1389:
1385:
1383:
1379:
1377:
1374:
1373:
1363:
1354:
1324:
1299:
1282:
1281:
1277:
1269:
1265:
1261:
1257:
1248:
1243:
1242:
1213:
1205:
1201:
1200:
1196:
1187:
1182:
1181:
1152:
1144:
1140:
1139:
1135:
1133:
1130:
1129:
1123:
1117:
1114:
1108:
1107:of the surface
1105:
1098:
1089:
1072:
1063:
1060:
1054:
1051:
1042:
1039:
1033:
1006:
1001:
1000:
971:
963:
959:
958:
954:
945:
940:
939:
910:
902:
898:
897:
893:
884:
879:
878:
849:
841:
837:
836:
832:
821:
818:
817:
804:
801:
795:
792:
786:
783:
777:
774:
768:
765:
759:
756:
750:
723:
716:
711:
710:
690:
682:
678:
677:
673:
664:
659:
658:
629:
621:
617:
616:
612:
611:
609:
594:
577:
570:
566:
562:
560:
558:
555:
554:
548:
542:
539:
533:
530:
524:
523:to the surface
520:
514:
513:is advected by
510:
507:
501:
495:
489:
483:
477:
474:
468:
465:
459:
421:
413:
412:
408:
389:
383:
379:
377:
375:
372:
371:
359:dominates over
349:
321:
288:
281:
277:
273:
271:
269:
266:
265:
256:
236:
230:
227:
204:
198:
195:
189:
186:
180:
177:
171:
164:
154:
124:
111:
68:
37:magnetic fields
17:
12:
11:
5:
3535:
3525:
3524:
3508:
3507:
3454:Lazarian, Alex
3444:
3407:
3354:
3327:(2): 177–197.
3311:
3258:
3251:
3220:
3167:
3154:978-9868270954
3153:
3130:
3093:(2): 489–494.
3077:
3070:
3049:
3042:
3016:
2977:
2934:
2931:. 29B(2): 1–7.
2911:
2910:
2908:
2905:
2902:
2901:
2884:
2883:
2881:
2878:
2877:
2876:
2871:
2866:
2859:
2856:
2852:Wiener process
2836:
2835:
2824:
2821:
2818:
2813:
2808:
2803:
2800:
2795:
2792:
2789:
2786:
2783:
2780:
2775:
2770:
2765:
2760:
2755:
2750:
2719:
2716:
2692:
2691:
2670:
2667:
2663:
2657:
2653:
2647:
2644:
2641:
2638:
2634:
2626:
2622:
2618:
2615:
2612:
2608:
2602:
2598:
2592:
2589:
2586:
2563:
2560:
2542:
2539:
2522:
2521:
2510:
2507:
2503:
2499:
2495:
2491:
2488:
2485:
2482:
2476:
2473:
2467:
2463:
2412:
2409:
2405:
2404:
2393:
2390:
2387:
2383:
2379:
2375:
2370:
2365:
2361:
2357:
2352:
2348:
2345:
2342:
2336:
2333:
2327:
2323:
2316:
2266:
2265:
2252:
2248:
2245:
2242:
2238:
2234:
2231:
2228:
2222:
2219:
2213:
2209:
2206:
2141:
2140:
2129:
2125:
2120:
2116:
2113:
2108:
2104:
2097:
2093:
2089:
2084:
2080:
2074:
2067:
2064:
2060:
2044:
2043:
2032:
2028:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1996:
1992:
1989:
1983:
1980:
1975:
1972:
1955:
1954:
1943:
1940:
1936:
1932:
1929:
1926:
1922:
1918:
1914:
1910:
1907:
1904:
1900:
1896:
1893:
1889:
1885:
1882:
1879:
1875:
1871:
1868:
1862:
1859:
1853:
1849:
1824:
1821:
1820:
1819:
1808:
1805:
1799:
1796:
1789:
1785:
1781:
1764:
1763:
1752:
1747:
1742:
1737:
1734:
1730:
1725:
1720:
1716:
1712:
1707:
1703:
1700:
1697:
1691:
1688:
1682:
1678:
1671:
1663:
1659:
1654:
1650:
1644:
1641:
1634:
1630:
1626:
1605:
1604:
1589:
1585:
1581:
1578:
1574:
1570:
1567:
1564:
1560:
1556:
1552:
1547:
1539:
1535:
1531:
1527:
1523:
1518:
1513:
1508:
1505:
1499:
1496:
1491:
1488:
1485:
1481:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1455:
1444:
1440:
1435:
1429:
1426:
1423:
1420:
1416:
1412:
1406:
1403:
1396:
1392:
1388:
1382:
1381:
1359:
1343:
1342:
1331:
1327:
1323:
1320:
1316:
1312:
1309:
1306:
1302:
1298:
1295:
1292:
1285:
1280:
1272:
1268:
1264:
1260:
1256:
1251:
1246:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1216:
1208:
1204:
1199:
1195:
1190:
1185:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1155:
1147:
1143:
1138:
1121:
1112:
1103:
1070:
1058:
1049:
1037:
1026:
1025:
1014:
1009:
1004:
999:
996:
993:
990:
987:
984:
981:
978:
974:
966:
962:
957:
953:
948:
943:
938:
935:
932:
929:
926:
923:
920:
917:
913:
905:
901:
896:
892:
887:
882:
877:
874:
871:
868:
865:
862:
859:
856:
852:
844:
840:
835:
831:
828:
825:
799:
790:
781:
772:
763:
754:
747:
746:
735:
729:
726:
719:
714:
709:
706:
703:
700:
697:
693:
685:
681:
676:
672:
667:
662:
657:
654:
651:
648:
645:
642:
639:
636:
632:
624:
620:
615:
606:
603:
600:
597:
593:
589:
583:
580:
573:
569:
565:
546:
537:
528:
505:
481:
472:
463:
445:
444:
433:
429:
424:
420:
416:
411:
407:
404:
401:
395:
392:
386:
382:
348:
345:
318:
317:
306:
303:
300:
294:
291:
284:
280:
276:
254:velocity field
232:
226:
223:
202:
193:
184:
175:
153:
150:
122:
110:
107:
67:
64:
56:current sheets
31:, states that
15:
9:
6:
4:
3:
2:
3534:
3523:
3520:
3519:
3517:
3503:
3499:
3494:
3489:
3485:
3481:
3476:
3471:
3468:(2): 025001.
3467:
3463:
3459:
3455:
3448:
3440:
3436:
3431:
3426:
3423:(8): 083102.
3422:
3418:
3411:
3403:
3399:
3395:
3391:
3387:
3383:
3378:
3373:
3370:(5): 056405.
3369:
3365:
3358:
3350:
3346:
3342:
3338:
3334:
3330:
3326:
3322:
3315:
3307:
3303:
3299:
3295:
3291:
3287:
3282:
3277:
3273:
3269:
3262:
3254:
3248:
3244:
3240:
3236:
3235:
3227:
3225:
3216:
3212:
3208:
3204:
3200:
3196:
3191:
3186:
3182:
3178:
3171:
3156:
3150:
3143:
3142:
3134:
3126:
3122:
3118:
3114:
3110:
3106:
3101:
3096:
3092:
3088:
3081:
3073:
3071:0-521-48179-1
3067:
3063:
3056:
3054:
3045:
3039:
3035:
3031:
3027:
3020:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2981:
2973:
2969:
2965:
2961:
2957:
2953:
2950:(3805): 405.
2949:
2945:
2938:
2930:
2923:
2916:
2912:
2896:
2889:
2885:
2875:
2872:
2870:
2867:
2865:
2862:
2861:
2855:
2853:
2849:
2819:
2806:
2801:
2798:
2793:
2790:
2787:
2781:
2778:
2758:
2748:
2741:
2740:
2739:
2737:
2733:
2724:
2715:
2713:
2709:
2704:
2698:
2655:
2642:
2616:
2600:
2587:
2577:
2576:
2575:
2573:
2572:magnetic flux
2569:
2559:
2557:
2553:
2549:
2538:
2535:
2529:
2508:
2497:
2486:
2480:
2474:
2451:
2450:
2449:
2446:
2440:
2436:
2435:
2428:
2424:
2418:
2408:
2391:
2388:
2385:
2377:
2373:
2368:
2359:
2350:
2346:
2340:
2334:
2314:
2306:
2305:
2304:
2301:
2299:
2294:
2291:
2285:
2282:
2276:
2272:
2240:
2232:
2226:
2220:
2217:
2207:
2204:
2194:
2193:
2192:
2189:
2185:
2182:
2178:
2175:
2169:
2166:
2162:
2156:
2150:
2147:
2127:
2118:
2111:
2106:
2095:
2091:
2087:
2082:
2072:
2065:
2062:
2058:
2049:
2048:
2047:
2030:
2022:
2016:
2013:
2010:
2007:
1998:
1987:
1981:
1973:
1960:
1959:
1958:
1941:
1930:
1916:
1902:
1891:
1877:
1866:
1860:
1837:
1836:
1835:
1833:
1828:
1806:
1803:
1797:
1794:
1787:
1779:
1769:
1768:
1767:
1750:
1745:
1735:
1732:
1728:
1723:
1714:
1705:
1701:
1695:
1689:
1669:
1661:
1657:
1652:
1648:
1642:
1639:
1632:
1624:
1614:
1613:
1612:
1610:
1587:
1579:
1576:
1572:
1565:
1554:
1545:
1537:
1533:
1525:
1521:
1516:
1506:
1503:
1497:
1494:
1486:
1475:
1469:
1466:
1463:
1460:
1442:
1438:
1433:
1427:
1421:
1418:
1410:
1404:
1401:
1394:
1386:
1372:
1371:
1370:
1367:
1362:
1357:
1352:
1348:
1329:
1321:
1318:
1314:
1307:
1296:
1293:
1290:
1278:
1270:
1266:
1258:
1254:
1249:
1239:
1236:
1230:
1227:
1224:
1221:
1206:
1202:
1197:
1193:
1188:
1178:
1175:
1169:
1166:
1163:
1160:
1145:
1141:
1136:
1128:
1127:
1126:
1120:
1111:
1102:
1095:
1092:
1086:
1083:
1079:
1076:
1069:
1066:
1057:
1048:
1045:
1036:
1031:
1012:
1007:
997:
994:
988:
985:
982:
979:
964:
960:
955:
951:
946:
936:
933:
927:
924:
921:
918:
903:
899:
894:
890:
885:
875:
872:
866:
863:
860:
857:
842:
838:
833:
829:
826:
823:
816:
815:
814:
811:
807:
798:
789:
780:
771:
762:
753:
733:
727:
724:
717:
707:
704:
698:
683:
679:
674:
670:
665:
655:
652:
646:
643:
640:
637:
622:
618:
613:
604:
598:
595:
587:
581:
578:
571:
563:
553:
552:
551:
545:
536:
527:
517:
504:
498:
492:
480:
471:
462:
456:
452:
450:
431:
427:
418:
409:
405:
399:
393:
370:
369:
368:
366:
362:
358:
354:
344:
342:
336:
332:
328:
324:
304:
301:
298:
292:
289:
282:
274:
264:
263:
262:
259:
255:
251:
248:
245:
241:
235:
222:
218:
216:
211:
201:
192:
183:
174:
168:
163:
159:
149:
147:
143:
138:
135:
134:length scales
131:
127:
125:
116:
106:
103:
101:
100:
93:
91:
90:eddy currents
85:
83:
82:
77:
76:Hannes Alfvén
73:
63:
61:
57:
53:
48:
46:
45:Hannes Alfvén
42:
38:
35:and embedded
34:
30:
26:
22:
3465:
3461:
3447:
3420:
3416:
3410:
3367:
3363:
3357:
3324:
3320:
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